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\author{Class 2019 Math and Applied Math }
\title{Applied stochastic processes - Homework 05}
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%\date{2021 年 2 月 28 日}
\date{April 20, 2021}

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%\subsection{Homework 05}
%E5.3.1, E5.3.3, E5.3.7, P5.3.1, E5.4.1, E5.4.3. 

\begin{document}

\maketitle

\begin{enumerate}

\item [E5.3.1.] A radioactive source emits particles according to a Poisson process of rate $\lambda=2$ particles per minute. 
What is the probability that the first particle appears after 3 min?


\item [E5.3.3.] Customers enter a store according to a Poisson process of rate $\lambda = 6$ per hour. 
Suppose it is known that only a single customer entered during the first hour. 
What is the conditional probability that this person entered during the first 15 min?


\item [E5.3.7.] Customers arrive at a service facility according to a Poisson process of rate $\lambda$ customers per hour. 
Let $X(t)$ be the number of customers that have arrived up to time $t$. 
Let $W_1$, $W_2$, $\cdots$ be the successive arrival times of the customers. 
Determine the conditional mean $$\mathbb{E} [ W_5 \mid X(t) = 3 ]. $$


\item [P5.3.1.] Let $X(t)$ be a Poisson process of rate $\lambda$. 
\begin{enumerate}

\item  Validate the identity $\{W_1 > w_1,W_2 > w_2\}$ if and only if 
\begin{eqnarray*}
\{X(w_1) = 0, X(w_2) - X(w_1) = 0 \text{ or } 1\}.\
\end{eqnarray*}

\item  Use (a) to determine the joint upper tail probability
\begin{eqnarray*}
\mathbb{P}\{W_1 > w_1, W_2 > w_2 \} 
&=& \mathbb{P} \{ X(w_1)=0, X(w_2) - X(w_1)=0 \text{ or } 1\} \\
&=&  e^{-\lambda w_1} [ 1 + \lambda (w_2 - w_1)] e^{-\lambda (w_2-w_1)}.
\end{eqnarray*}

\item  Finally, differentiate twice to obtain the joint density function 
\begin{eqnarray*}
f(w1,w2) = \lambda^2 \exp(-\lambda w_2) \text{ for }  0 < w_1 < w_2.
\end{eqnarray*}

\end{enumerate}


\item  [E5.4.1.] Let $\{X(t);t\ge 0\}$ be a Poisson process of rate $\lambda$. 
Suppose it is known that $X(1) = n$. For $n = 1,2,\cdots$, determine the mean of the first arrival time $W_1$.


\item [E5.4.3.] Customers arrive at a certain facility according to a Poisson process of rate $\lambda$. 
Suppose that it is known that five customers arrived in the first hour. 
Determine the mean total waiting time $\mathbb{E} [ W_1 + W_2 + \cdots + W_5 ]$. 



\end{enumerate}


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\subsection{Homework 01}
E3.1.2, P3.1.4, E3.2.2, P3.2.4, E3.3.2, P3.3.6.

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\subsection{Homework 02}
E.3.4.1, E3.4.2, P3.4.1, P3.4.5, E3.5.1, P3.5.1. 

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\subsection{Homework 03}
E4.1.10, P4.1.1, P4.1.5, E4.3.1, E4.3.2, E4.4.2.

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\subsection{Homework 04}
E5.1.1, E5.1.7, P5.1.10, E5.2.1, P5.2.1.

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\subsection{Homework 05}
E5.3.1, E5.3.3, E5.3.7, P5.3.1, E5.4.1, E5.4.3. 

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\subsection{Homework 06}
E6.1.1, E6.1.2, P6.1.1, P6.1.2.

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\subsection{Homework 07}
E7.1.2, E7.1.3, E7.2.1, E7.2.3, P7.2.1.

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\subsection{Homework 08}
E8.1.1, E8.1.2, E8.1.4, P8.1.1, P8.1.3, E8.2.1.

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